We study the $H$-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group $H^n=C^n\times R\equiv R^{2n+1}$ endowed with a left-invariant metric $d_{\infty}$ equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words ``intrinsic'' and ``regular'' we mean respectively notions involving the group structure of $H^n$ and its differential structure as CC manifold. In particular we characterize these surfaces as intrinsic regular graphs inside $H^n$ by studying the intrinsic regularity of the parameterizations and giving an area-type formula for their intrinsic surface measure.